I think I am the only one who has no problem with the word "brat". I don't see Claire as calling the child a brat, as in what everyone else seems to be thinking, in that the child is a bad child.
I have a four year old granddaughter who I love dearly, and I call her "Bratella". But, she has never, ever been a brat. Well, for me at least, not her mommy! So, the nickname did not derive from being a bad kid, just my little cutie patootie. Actually, I think my mom came up with the nickname.
I saw it as Claire saying the word brat instead of kid, but not meaning bad child, just a child.
And le'ts be realistic here, going by the parents themselves and already how they are acting, who here who has posted doesn't think that this kid growing up is going to be one, big, spoiled diva??? (or, ok, a brat?).